and Cosmology

4.4 Thermal History of the Universe
photons by (11/4) 1/3 ∼ 1.4 – until the present epoch.
This result has already been mentioned and taken into
account in the estimate of ρ r,0 in (4.26); we find
ρ r,0 = 1.68ρ CMB,0 .
After pair annihilation, the expansion law
( ) T −2
t = 0.55 s
(4.58)
1MeV
applies. This means that, as a result of annihilation, the
constant in this relation changes compared to (4.56)
because the number of relativistic particles species has
decreased. Furthermore, the ratio η of baryon to photon
density remains constant after pair annihilation. The
former is characterized by the density parameter Ω b =
ρ b,0 /Ω cr in baryons (today), and the latter is determined
by T 0 :
( )
nb
η := = 2.74 × 10 −8 ( Ω b h 2) . (4.59)
n γ
Before pair annihilation there were about as many
electrons and positrons as there were photons. After
annihilation nearly all electrons were converted into
photons – but not entirely because there was a very
small excess of electrons over positrons to compensate
for the positive electrical charge density of the protons.
Therefore, the number density of electrons that survive
the pair annihilation is exactly the same as the number
density of protons, for the Universe to remain electrically
neutral. Thus, the ratio of electrons to photons is
also given by η, or more precisely by about 0.8η, since
η includes both protons and neutrons.
4.4.4 Primordial Nucleosynthesis
Protons and neutrons can fuse to form atomic nuclei
if the temperature and density of the plasma are sufficiently
high. In the interior of stars, these conditions for
nuclear fusion are provided. The high temperatures in
the early phases of the Universe suggest that atomic
nuclei may also have formed then. As we will discuss
below, in the first few minutes after the Big Bang
some of the lightest atomic nuclei were formed. The
quantitative discussion of this primordial nucleosynthesis
(Big Bang nucleosynthesis, BBN) will explain
observation (4) of Sect. 4.1.1.
Proton-to-Neutron Abundance Ratio. As already discussed,
the baryons (or nucleons) do not play any role in
the expansion dynamics in the early Universe because of
their low density. The most important reactions through
which they maintain chemical equilibrium with the rest
of the particles are
p + e ↔ n + ν,
p + ν ↔ n + e + ,
n → p + e + ν.
The latter is the decay of free neutrons, with a timescale
for the decay of τ n = 887 s. The first two reactions
maintain the equilibrium proton-to-neutron ratio as long
as the corresponding reaction rates are large compared
to the expansion rate. The equilibrium distribution is
specified by the Boltzmann factor,
)
n n
= exp
(− Δmc2 , (4.60)
n p k B T
where Δm = m n − m p = 1.293 MeV/c 2 is the mass difference
between neutrons and protons. Hence, neutrons
are slightly heavier than protons; otherwise the neutron
decay would not be possible. After neutrino freeze-out
equilibrium reactions become rare because the above reactions
are based on weak interactions, the same as those
which kept the neutrinos in chemical equilibrium. At
the time of neutrino decoupling, we have n n /n p ≈ 1/3.
After this, protons and neutrons are no longer in equilibrium,
and their ratio is no longer described by (4.60).
Instead, it changes only by the decay of free neutrons
on the time-scale τ n . To have neutrons survive at all until
the present day, they must quickly become bound in
atomic nuclei.
Deuterium Formation. The simplest compound nucleus
is that of deuterium (D), consisting of a proton
and a neutron and formed in the reaction
p + n → D + γ.
The binding energy of D is E b = 2.225 MeV. This energy
is only slightly larger than m e c 2 and Δm –all
these energies are of comparable size. The formation
of deuterium is based on strong interactions and therefore
occurs very efficiently. However, at the time of
neutrino decoupling and pair annihilation, T is not
much smaller than E b . This has an important consequence:
because photons are so much more abundant
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